## On multivariate Lagrange interpolation

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- by Thomas Sauer and Yuan Xu PDF
- Math. Comp.
**64**(1995), 1147-1170 Request permission

## Abstract:

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree*n*of a function

*f*, which is a sum of integrals of certain $(n + 1)$st directional derivatives of

*f*multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

## References

- Carl de Boor and Amos Ron,
*Computational aspects of polynomial interpolation in several variables*, Math. Comp.**58**(1992), no. 198, 705–727. MR**1122061**, DOI 10.1090/S0025-5718-1992-1122061-0 - M. Gasca,
*Multivariate polynomial interpolation*, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 215–236. MR**1064962**, DOI 10.1098/rspa.1968.0185 - Eugene Isaacson and Herbert Bishop Keller,
*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039** - Rudolph A. Lorentz,
*Multivariate Birkhoff interpolation*, Lecture Notes in Mathematics, vol. 1516, Springer-Verlag, Berlin, 1992. MR**1222648**, DOI 10.1007/BFb0088788 - Charles A. Micchelli,
*On a numerically efficient method for computing multivariate $B$-splines*, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 211–248. MR**560673**

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp.
**64**(1995), 1147-1170 - MSC: Primary 41A63; Secondary 41A05, 65D05
- DOI: https://doi.org/10.1090/S0025-5718-1995-1297477-5
- MathSciNet review: 1297477